Quasi-Uniform Codes and Their Applications

Quasi-uniform random vectors have probability distributions that are uniform over their projections. They are of fundamental interest because a linear information inequality is valid if and only if it is satisfied by all quasi-uniform random vectors. In this paper, we investigate properties of codes induced by quasi-uniform random vectors. We prove that quasi-uniform codes (which include linear and almost affine codes as special cases) are distance-invariant and that Greene's Theorem and the Critical Theorem of Crapo and Rota hold in the setting of quasi-uniform codes. We show that both theorems are essentially combinatorial but not algebraical in nature. Linear programming bounds proposed by Delsarte are extended for quasi-uniform codes.